We consider a robust version of the full information best choice
problem (Gilbert and Mosteller (1966)): there is ambiguity (represented
by a set of priors) about the measure driving the observed
process. We solve the problem under a very general class of multiple
priors in the setting of Riedel (2009). As in the classical case, it is
optimal to stop if the current observation is a running maximum that
exceeds certain thresholds. We characterize the decreasing sequence
of thresholds, as well as the (history dependent) minimizing measure.
We introduce locally constant ambiguity neighborhood (LCAn) which
has connections to coherent risk measures. Sensitivity analysis is performed
using LCAn and exponential neighborhood from Riedel (2009).